# The angles subtended in a TSP tour

If I sample a large number of uniform points in the unit square and take a traveling salesman tour of them, is there anything at all that can be said/is known about the distribution of angles at each point? The two pictures below show a TSP tour of 1000 points, together with a histogram of the angles.

It is intuitive that we expect a lot of angles to be near $\pi$, since an angle close to $0$ (or to $2\pi$) suggests that we're "doubling back" on a route and possibly being inefficient.

• A further question would be how the angles between entering and leaving a sequence of left (resp. right) turns is distributed. As there is a difference of $2\pi$ between cumulated left and right turns, separate statistics may provide further "insights". Apr 2, 2017 at 6:59
• Just a tangential remark: There is work on the Dubins' TSP: visiting the points by a vehicle that can only move forward and has a limited turning radius. In some sense, constraining the angle histogram. Apr 3, 2017 at 0:49

The distribution has a stunning similarity to the density function of random Delaunay angles mentioned on page 4 of The Expected Extremes in a Delaunay Triangulation

An explanation might be, that optimal tours go along Delaunay edges for a major part of their course (that heuristic made me search for properties of Delaunay triangulations in the first place).